We define two numerical operations labeled T and R.
The effect of T is to add 1 to a number. For example, if we apply the operation T to the number 2 three times in a row, we obtain 3, then 4, then 5.
The effect of R is to find the negative reciprocal of a number. For example, if we apply the operation R to 2 we obtain – ½, and if we apply the operation R to -3/2 we obtain 2/3. Note that R can never be applied to the number 0.
The operations T and R can be combined. For example, we can turn 0 into 2/5 by successively applying the operations T, T, T, R, T, T, R, T:
0 ---T ---> 1 ---T---> 2 ---T ---> 3 ---R---> -1/3 ---T ---> 2/3 ---T ---> 5/3 ---R ---> -3/5 ---T ---> 2/5.
a. Starting with 2, list the numbers produced by successively applying the operations
T, R, R, T, R, T, R, T, R.
b. Find a sequence of operations which turns ¾ into 2/3
c. Find a sequence of operations which turns 3 into 0.
d. Find a sequence of 20 operations that turn 7 into 0.
Please help
a. 3 , -1/3 , 3 , 4 , -1/4 , 3/4 , -4/3 , -1/3 , 3
b. R , T , T
c. R , T , R , T , T , R , T , T
d. R , T , R , T , T , R , T , T , R , T , T , R , T , T , R , T , T , R , T , T
scott is correct although I don't think B is.
ohhh nevermind, I understand that scott was doing. my bad lol
This is in a comp booklet, why did you post this???
how does a go from -1/4 to 3/4??
ohh i get it know tho
Sure, I can help you with that.
a. To list the numbers produced by successively applying the operations T, R, R, T, R, T, R, T, R starting with 2, we will follow these steps:
1. Start with the number 2.
2. Apply operation T once to get 3.
3. Apply operation R twice to get -1/3.
4. Apply operation T once to get 2/3.
5. Apply operation R once to get -3/2.
6. Apply operation T once to get -1/2.
7. Apply operation R once to get 2.
8. Apply operation T once to get 5/2.
9. Apply operation R once to get -2/5.
So the sequence of numbers produced by these operations starting with 2 is 2, 3, -1/3, 2/3, -3/2, -1/2, 2, 5/2, -2/5.
b. To find a sequence of operations which turns 3/4 into 2/3, we will follow these steps:
1. Start with the number 3/4.
2. Apply operation R once to get -4/3.
3. Apply operation T once to get -1/3.
4. Apply operation R once to get 3.
5. Apply operation T once to get 4.
6. Apply operation R once to get -1/4.
7. Apply operation T once to get 3/4.
8. Apply operation R once to get -4/3.
9. Apply operation T once to get -1/3.
10. Apply operation R once to get 3/4.
11. Apply operation T once to get 4/4, which simplifies to 1.
So the sequence of operations that turns 3/4 into 2/3 is R, T, R, T, R, T, R, T, R, T.
c. To find a sequence of operations which turns 3 into 0, we will follow these steps:
1. Start with the number 3.
2. Apply operation R once to get -1/3.
3. Apply operation T once to get 2/3.
4. Apply operation R once to get -3/2.
5. Apply operation T once to get -1/2.
6. Apply operation R once to get 2.
7. Apply operation T once to get 5/2.
8. Apply operation R once to get -2/5.
9. Apply operation T once to get 5/5, which simplifies to 1.
10. Apply operation R once to get -1.
11. Apply operation T once to get 0.
So the sequence of operations that turns 3 into 0 is R, T, R, T, R, T, R, T, R, T, R, T.
d. To find a sequence of 20 operations that turn 7 into 0, we will follow these steps:
1. Start with the number 7.
2. Apply operation R once to get -1/7.
3. Apply operation T once to get 6/7.
4. Apply operation R once to get -7/6.
5. Apply operation T once to get -5/6.
6. Apply operation R once to get 6/5.
7. Apply operation T once to get 11/5.
8. Apply operation R once to get -5/11.
9. Apply operation T once to get 6/11.
10. Repeat steps 2-9 nine more times to get the following sequence: -1/7, 6/7, -7/6, -5/6, 6/5, 11/5, -5/11, 6/11, -11/17, 6/17, -17/23, 6/23, -23/29, 6/29, -29/35, 6/35, -35/41, 6/41, -41/47, 6/47.
11. Apply operation R once to get -47/6.
12. Apply operation T once to get -41/6.
13. Apply operation R once to get 6/41.
14. Apply operation T once to get 47/41.
15. Apply operation R once to get -41/47.
16. Apply operation T once to get 6/47.
17. Apply operation R once to get 47/6.
18. Apply operation T once to get 41/6.
19. Apply operation R once to get -6/41.
20. Apply operation T once to get 47/41.
So the sequence of 20 operations that turn 7 into 0 is R, T, R, T, R, T, R, T, R (repeated 9 times), R, T, R, T, R, T, R, T, R, T.